This journal is©The Royal Society of Chemistry 2016 Soft Matter

Cite this:DOI: 10.1039/c6sm01955b

Geometric control of active collective motion†

Maxime Theillard, Roberto Alonso-Matilla and David Saintillan*

Recent experimental studies have shown that confinement can profoundly affect self-organization in semi-

dilute active suspensions, leading to striking features such as the formation of steady and spontaneous

vortices in circular domains and the emergence of unidirectional pumping motions in periodic racetrack

geometries. Motivated by these findings, we analyze the two-dimensional dynamics in confined suspensions

of active self-propelled swimmers using a mean-field kinetic theory where conservation equations for

the particle configurations are coupled to the forced Navier–Stokes equations for the self-generated

fluid flow. In circular domains, a systematic exploration of the parameter space casts light on three

distinct states: equilibrium with no flow, stable vortex, and chaotic motion, and the transitions between

these are explained and predicted quantitatively using a linearized theory. In periodic racetracks, similar

transitions from equilibrium to net pumping to traveling waves to chaos are observed in agreement with

experimental observations and are also explained theoretically. Our results underscore the subtle effects

of geometry on the morphology and dynamics of emerging patterns in active suspensions and pave the

way for the control of active collective motion in microfluidic devices.

1 IntroductionA common feature of many active matter systems is their abilityto spontaneously self-organize into complex dynamic mesoscalestructures above a certain density.1–3 Such is the case of suspen-sions of motile bacteria,4–7 cellular extracts,8–10 collections ofcolloidal rollers,11,12 shaken grains,13,14 among many others.Particle–particle interactions, whether long-ranged such as hydro-dynamic or electrostatic interactions, or short-ranged such asdirect contact forces, are the drivers of self-organization.1

The symmetries of these interactions along with their couplingwith system geometry dictates the structure and morphology ofthe emerging patterns, which include: steady vortices,12,15

asters,8 traveling bands,9,11 density shocks,16,17 as well as morecomplex spatiotemporal chaotic patterns composed of unsteadyjets and vortices.4

Of interest to us in this work is the case of suspensions ofhydrodynamically interacting slender self-propelled particlessuch as swimming bacteria.18 In these suspensions, particlesexert dipolar stresses on the surrounding medium and also alignin shear due to their elongated shape. The interplay betweenthese two effects has been known to lead to hydrodynamicinstabilities in the case of extensile particles or so-called pushers,which are thought to be responsible for the emergence of

collective motion above a critical density.19–22 In large unconfinedsystems, the collective dynamics in the nonlinear regime takes theform of unsteady chaotic motions reminiscent of high-Reynolds-number turbulence, characterized by strong jets and vortices,enhanced swimming speeds and diffusivities, and efficient fluidmixing.23

Only recently have interactions with boundaries and dynamicsin confined geometries gained attention in experiments. In dilutesystems, it is well known that self-propelled particles accumulateat boundaries24–27 as a result of both kinematic25,28–31 andhydrodynamic mechanisms.24,32,33 In complex geometries,transport of the particles along curved boundaries has also beenexploited to design ratchets for concentrating microswimmers ordirecting their motion.34–38 The case of semi-dilute to concen-trated suspensions in confinement, however, has largely beenunexplored but in a few studies. Wioland et al.39 first analyzedthe flow inside small droplets of a dense bacterial suspensionsqueezed between two flat plates. Rather than observing chaoticmotion as in bulk systems, they reported the emergence ofa steady vortex; detailed observation of the bacterial velocityfield in fact revealed a more complex structure with a counter-rotating boundary layer surrounding the vortex core. Thisvortex was subsequently captured by Lushi et al.40 in discreteparticle simulations using a basic model accounting for dipolarhydrodynamic interactions as well as steric forces, where itwas found that including hydrodynamic interactions is criticalin order to correctly capture the counter-rotating boundarylayer. Interactions between such vortices were also consideredrecently using a microfluidic lattice of circular chambers each

Department of Mechanical and Aerospace Engineering, University of CaliforniaSan Diego, 9500 Gilman Drive, La Jolla CA 92093, USA. E-mail: dstn@ucsd.edu;Tel: +1 858-822-7925† Electronic supplementary information (ESI) available: Movies showing simulationresults in various geometries and flow regimes. See DOI: 10.1039/c6sm01955b

Received 24th August 2016,Accepted 18th November 2016

DOI: 10.1039/c6sm01955b

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containing one vortex and connected by junctions:41 in thiscase, hydrodynamic coupling was shown to produce synchro-nization on the scale of the lattice, with adjacent vorticesrotating either in the same or opposite direction dependingon the geometry of the junctions between chambers.

The case of periodic geometries such as circular channels andracetracks has also been studied, where spontaneous flows havebeen reported in both bacterial42 and sperm43 suspensions above acritical density. In the case of bacteria, Wioland et al.42 systematicallystudied the effect of geometry by varying the channel width. In verynarrow channels, unidirectional flow takes place with a nearlyparabolic velocity profile. Upon increasing channel width, flowpatterns start exhibiting longitudinal oscillations leading to sinusoi-dal trajectories and eventually take the form of arrays of counter-rotating vortices. Longitudinal density waves were also reported inthe case of dense semen.43 The observed transition to directedmotion has been predicted in a number of models for activenematics44,45 and active polar fluids,46,47 where extensile stresseswere found to be the destabilizing factor leading to spontaneousflows. These models, however, either neglected polarization,44,45

which plays an important role in setting the structure of thesuspension in confined systems of self-propelled particles,30 ornematic alignment,46,47 which is responsible for driving activestresses; they also assumed anchoring boundary conditions for thenematic order parameter field at the channel boundaries, whereasthe distribution of particle orientations near the walls appears to bedependent on flow conditions in experiments.39,42 A qualitativeexplanation for the transition can also be gleaned from recentstudies on the effective rheology of active suspensions,48–51 wherea decrease of the effective viscosity due to activity can lead atsufficiently high densities to a superfluid-like behavior in weakflows; this connection will be made clearer below.

In this paper, we use numerical simulations based on acontinuum kinetic model together with linear stability analysesto predict and characterize transitions to spontaneous flows andcollective motion in various two-dimensional microfluidic geo-metries, with the aim of explaining the experimental observationsdiscussed above. A distinguishing feature of our model compared toprevious theories44–47 is its focus on hydrodynamic interactions,which, as we demonstrate below, are sufficient to explain all thephenomenology observed in experiments with no need for stericinteractions. The governing equations are presented in Section 2 andconsist of evolution equations for the concentration, polarizationand nematic order parameter, which are coupled to the Navier–Stokes equations for the mean-field flow induced by the swimmers.Results from simulations and theory are then discussed in Section 3,where both circular domains and periodic racetracks are considered.We conclude in Section 4.

2 Model and simulation method2.1 Continuum model

We consider a collection of active particles suspended in aNewtonian fluid of density r and shear viscosity m. The particlesswim with velocity Vs and have constant translational and

rotational diffusivities dt and dr, respectively. As a result of theirself-propulsion, they also exert a net force dipole on the suspendingfluid with stresslet strength s0, which we assume to be negative asis the case for extensile swimmers such as bacteria and sperm.52,53

The suspension, with mean number density n, is confined in afinite domain with characteristic dimension H, which will bedefined more precisely later. Dimensional analysis of the governingequations identifies four relevant dimensionless groups:

Re ¼ rH2drm

; Pes ¼Vs

drH; a ¼ s0n

mdr; L ¼ drdt

Vs2: (1)

The Reynolds number Re is typically very small for activesuspensions; it will be set to 10"6 in all the simulations and tozero in the stability results shown below. The swimming Pecletnumber Pes denotes the ratio of the persistence length ofswimmer trajectories to the size of the domain and is a measureof confinement. The activity parameter a compares the destabi-lizing effects of active stresses and of concentration to dissipativeprocesses, namely viscosity and orientation decorrelation byrotational diffusion. Finally, L is a swimmer-specific parametercomparing diffusive processes to the strength of self-propulsion:the limit of L- 0 describes athermal swimmers, whereas L-Ncorresponds to particles that diffuse but do not swim.

We adopt a two-dimensional continuum mean-field descriptionof the active suspension based on the probability density functionC(x,p,t) of finding a particle at position x with orientation p attime t, where p is a unit vector defining the swimming directionand orientation of the bacteria.19 Following prior studies,2 weapproximate C in terms of its first three orientational moments:

Cðx; p; tÞ % 1

2pcðx; tÞ þ 2p 'mðx; tÞ þ 4pp:Dðx; tÞ½ ); (2)

where c, m, and D are defined as integrals over the unit circle Cof orientations:

cðx; tÞ ¼ð

CCðx; p; tÞdp; (3)

mðx; tÞ ¼ð

CpCðx; p; tÞdp; (4)

Dðx; tÞ ¼ð

Cpp" I

2

" #Cðx; p; tÞdp: (5)

The zeroth moment c is the local concentration, whereas the firstand second moments m and D describe the local polarization andnematic alignment in the suspension, respectively. Starting from aSmoluchowski equation for C(x,p,t),19 hierarchical evolutionequations for the moments can be obtained, which are written as:

qtc = "r'Fc, (6)

@tm ¼ "r ' Fm þ1

2zE 'm"W 'm"m; (7)

@tD ¼ "r ' FD þ1

2zcEþD 'W"W 'D" 4D; (8)

where the dimensionless shape parameter z denotes Bretherton’sconstant;54 we set z = 1 in the present study as is adequate for

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slender swimmers. Variables have been made dimensionlessusing length scale H and time scale dr

"1. The source terms onthe right-hand side of eqn (7) and (8) arise from alignment androtation by the rate-of-strain and vorticity tensors E and W ofthe disturbance velocity field u, and from rotational diffusionwhich promotes relaxation towards isotropy with m = 0 andD = 0. The fluxes in the equations for c, m, and D includecontributions from advection by the flow, self-propulsion andtranslational diffusion, and are given by

Fc = uc + Pesm " LPes2rc, (9)

Fm ¼ umþ Pes Dþ cI

2

" #" LPes2rm; (10)

FD ¼ uDþ Pes T"mI

2

" #" LPes2rD; (11)

where the third-order tensor T is the third orientationalmoment and is related to the polarization according to theclosure approximation implied by eqn (2) as

Tijk ¼1

4midjk þmjdik þmkdij$ %

: (12)

Direct steric interactions between swimmers are neglectedwithin this model. Their leading effect is expected to be anenhancement of local nematic alignment due to the elongatedshape of the particles; this effect could easily be incorporated inour model using a nematic alignment potential as previouslydone by Ezhilan et al.,55 though hydrodynamic interactionsalone are sufficient to capture all the phenomenology observedin experiments.

Finally, the disturbance fluid velocity field u satisfies theincompressible Navier–Stokes equations forced by the divergenceof the active stress tensor aD:

r'u = 0, Re(qtu + u'ru) = "rp + r2u + ar'D. (13)

Note that additional passive stresses also arise due to theinextensibility of the particles in the flow field they generate:we neglect those here as it can be shown that they only act toincrease the Newtonian viscosity in the limit of weak flowsrelevant to the spontaneous flow transitions investigated here,and therefore only renormalize the value of a at the instabilitythreshold. Including these stresses would be straightforwardand the reader is referred to our previous work50,51 on therheology of active suspensions for more details.

In all of our simulations and analysis, we enforce a no-slipboundary condition on the velocity u on the domain boundaryS. The natural boundary condition for the particle distribu-tions is a no-translational-flux condition on the probabilitydensity function C, which translates into no-flux conditionson the orientational moments upon closure of the equations:30

n'Fc = n'Fm = n'FD = 0, where n is the local unit normal. Theseconditions express the balance between self-propulsion andtranslational diffusion in the wall-normal direction, and wereshown to correctly capture particle distributions near boundariesin confined systems.30

2.2 Numerical approach

We solve the governing equations numerically using a hybridfinite-difference finite-volume framework.56–59 The method isimplemented on adaptive quadtree grids and the domainboundaries are represented using the level-set method. At eachtime step of the algorithm, the moment eqn (6)–(8) are solvedsemi-implicitly: the diffusive terms are treated implicitly andthe advective terms are computed using a semi-Lagrangianapproach for improved stability, whereas the remaining couplingterms are treated explicitly. Values of the concentration field arestored at the cell centers to improve its total conservation, whilethe polarization and nematic order parameter fields are stored atthe mesh nodes for better accuracy. Knowledge of the secondmoment D allows one to calculate the divergence of the activestress ar'D, which is an input to the Navier–Stokes solver59 usedto update the fluid velocity field. All the simulations shownhere are initialized with a uniform isotropic distribution (c = 1,m = 0, D = 0, u = 0). No initial perturbation is imposed in thesimulations, where minute numerical errors are instead sufficientto trigger instabilities and spontaneous flows. Even thoughour method generalizes to three spatial dimensions and theimplementation is straightforward, we limit the present studyto two spatial dimensions. A more detailed description of thealgorithm will be presented elsewhere together with resultsfrom three-dimensional simulations.

3 Results and discussion3.1 Circular disks

Motivated by the experiments of Wioland et al.39 in quasi-two-dimensional droplets, we first investigate the dynamicsin circular domains, where we take the confinement lengthscale H to be the radius of the disk. Our simulations in this caseshow that the collective self-organization depends criticallyon the level of activity (parameter a) and degree of confinement(swimming Peclet number Pes). Specifically, three distinctphases illustrated in Fig. 1 are observed depending on the valuesof a and Pes: an axisymmetric equilibrium state with no fluidflow (phase I), an axisymmetric and steady double vortex (phaseII), and a turbulent-like unsteady chaotic state (phase III). Insome cases, more complex axisymmetric flow patterns can alsobe observed before the transition to chaos, including stabletriple vortices as illustrated in Fig. 2; such flow patterns are onlyobserved for a very few parameter values and we do not discussthem further. Transitions between the three regimes, which arecharacterized in more detail below, can be captured in a phasediagram in the (a,Pes)-plane as shown in Fig. 3(a), where we findthat either increasing activity or decreasing confinement succes-sively destabilizes phase I into phase II followed by phase III.While distinguishing between these states is straightforward bysimple observation of the dynamics, we also introduce an orderparameter as a quantitative measure:

F ¼ 2

p

ð

O

Vyj jVj j

dx" 1

& '

t

; (14)

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where V = Pesm(x)/c(x) + u(x) is the net velocity of the activeparticles due to both swimming and advection by the flow, Vy

is its azimuthal component, O is the circular domain, and h'itdenotes a time average. Note that the velocity V is the same asthat measured in experiments, which typically perform particle-image velocimetry based on swimmer displacements.39,42 Theorder parameter in eqn (14) is defined such that F = "1 forpurely radial motion (phase I), F = +1 for purely azimuthal motion

(phase II), and F ¼ffiffiffi2p" 1 for a system in which motion occurs

equally along the radial and azimuthal directions (phase III).A plot of F vs. |a| for various values of Pes is shown in Fig. 3(b),where it indeed jumps from "1 to E+1 as the transition fromphase I to phase II occurs, before eventually decreasing to a value

close toffiffiffi2p" 1 as the chaotic state of phase III emerges. We

now proceed to characterize the three phases in more detail.Phase I: equilibrium state with no flow. This regime, illu-

strated in Fig. 1(a), occurs for low levels of activity (small |a|)and strong levels of confinement (large Pes), and is character-ized by the absence of hydrodynamic flow. As is known to bethe case in dilute confined active suspensions,30 particles tend

to accumulate near the system boundaries and on average pointtowards the boundary, which leads to a net radial polarization

Fig. 1 Concentration profiles and streamlines of the net particle velocity V(x) = Pesm(x)/c(x) + u(x) showing three distinct regimes in a circulardomain: (a) equilibrium base state with no flow (phase I); (b) double vortex flow (phase II); and (c) turbulent swirling state (phase III). Results shown arefor Pes = 1 and L = 0.1. Also see ESI† for a movie showing the dynamics in each case.

Fig. 2 Axisymmetric triple vortex in which the net particle velocity Vy

changes sign twice across the disk radius; this state is only rarely observedand therefore not included in the phase diagram of Fig. 3. (a) Concentrationprofile and streamlines of the net particle velocity. (b) Radial profile of theazimuthal particle velocity averaged over the azimuthal direction. Parametervalues for this simulation are Pes = 0.4, a = "40, and L = 0.2.

Fig. 3 Flow transitions in circular disks. (a) Phase diagram in the (a,Pes)-planefor L = 0.1 showing the transitions between phases I, II and III. The black curveshows the marginal stability for the equilibrium state of phase I as predicted bya linear stability analysis. (b) Order parameter F defined in eqn (14) as afunction of activity |a| for three different values of Pes. F = "1 correspondsto purely radial motion, F = +1 to purely azimuthal motion, and F ¼

ffiffiffi2p" 1

to equal amounts of radial and azimuthal motion.

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mr(r) 4 0 that reaches its maximum at the walls. The azimuthalpolarization my is zero in this case, as are off-diagonal compo-nents Dry and Dyr of the nematic order tensor. An analyticalsolution for the first two moments can in fact be derived inthis case by neglecting nematic alignment (D = 0), which is agood approximation as shown by full numerical simulations.The solution, given in Appendix A, matches the numericalresults excellently and predicts a dimensionless characteristicthickness for the wall accumulation layer given by

O"1 ¼ LPesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=ð1þ 2LÞ

p; (15)

which can also be interpreted as the length scale over which theeffects of the boundary are screened by diffusive processes.60 Asfirst discussed by Ezhilan & Saintillan,30 two interesting limitingcases are found. When L - 0 (strong-propulsion limit), thethickness of the layer is set by the balance of swimming and

translational diffusion and is given by O"1H %ffiffiffi2p

dt)Vs; on the

other hand, the weak-propulsion limit of L-N yields a thickness

of O"1H %ffiffiffiffiffiffiffiffiffiffiffidt=dr

p, which is a purely diffusive length scale.

Phase II: steady axisymmetric vortex. Upon decreasing con-finement or increasing the level of activity, a first transition isobserved from the equilibrium state to a steady axisymmetricdouble vortex shown in Fig. 1(b), as a consequence of thecoupled effect of nematic alignment induced by hydrodynamicinteractions and base-state heterogeneities in the concentrationand radial polarization profiles. The direction of rotation in thiscase is arbitrary with equal probabilities for clockwise andcounter-clockwise motions, though it is found to remain thesame for the duration of the simulation. More details on thisregime, which is identical to that reported in the experiments of

Wioland et al.,39 are shown in Fig. 4, where profiles of theazimuthal components of the fluid velocity uy, net particlevelocity Vy, polarization my, and active flow forcing fy = ar"2qr(r

2Dry)are plotted for different parameter values. While the azimuthalfluid velocity uy always points in the same direction, the netparticle velocity Vy follows the fluid flow near the center of thedomain but changes sign at a distance away from the boundary,indicating that particles near the circular wall move againstthe local flow in agreement with experiments and previousmodels.39,40 Fig. 4(c) in fact shows that the azimuthal polariza-tion is negative, i.e. particles swim against the flow everywherebut only overcome it near the wall. This observation is consis-tent with the well-known phenomenon of upstream swimming,by which near-wall bacteria tend to swim against the fluid inpressure-driven channel flows of active suspensions.30,61–64

As the transition from phase I to phase II occurs, we also findthat particle accumulation at the boundary is weakened, whichis a consequence of the alignment of the particles with thehydrodynamic flow, which reduces the wall-normal polariza-tion and therefore hinders the ability of the particles to swimtowards the wall; a similar effect is again also known to occur inpressure-driven channel flows.27,30

Mechanistic insight into the formation of the double vortexcan be gained by performing a linear stability analysis, in whichwe theoretically analyze the growth of axisymmetric perturbations

Fig. 4 Double vortex flow in a circular disk (phase II): radial profiles of (a)the azimuthal fluid velocity uy, (b) the azimuthal net particle velocity Vy, (c)the azimuthal polarization my, and (d) the active flow forcing fy =ar"2qr(r

2Dry) for different levels of activity, for Pes = 2 and L = 0.1.

Fig. 5 Linear stability of the equilibrium base state (phase I) in a circulardisk. (a) Marginal stability curves in the (a,Pes)-plane for the first threeunstable modes at L = 0.25. (b) Unstable eigenmodes for the net particlevelocity Vy for Pes = 1, and L = 0.25 at the onset of instability. (c) Marginalstability curves for the first unstable mode for different values of Lobtained by linear stability analysis (full lines); symbols show the marginalcurve for the transition from phase I to phase II in numerical simulations.

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to the equilibrium base state with no flow (phase I). Results fromthis analysis are summarized in Fig. 5. The analysis indeed revealsa linear instability of the equilibrium base state, with a hierarchyof unstable modes for which we plot the marginal stability curvesin the (a,Pes)-plane in Fig. 5(a) and the net azimuthal velocity Vy inFig. 5(b). The first unstable mode has a structure that is verysimilar to the nonlinear flow field of Fig. 4(b), with a vortex coreand a counter-rotating boundary layer near the domain wall.Subsequent modes, which become unstable at higher values of|a| as shown in Fig. 5(a), exhibit more and more complex structureswith alternating layers rotating in opposite directions. These modesare only very rarely observed in simulations, and an example ofa triple vortex similar for the second unstable mode is shown inFig. 2; in most cases, however, we find that the double vortexinstead destabilizes directly into the chaotic state.

The linearized theory also sheds light on the mechanism forthe transition, which can be summarized as follows: (i) at equili-brium (phase I), particles near the boundary have a net polariza-tion towards the wall;30 (ii) a weak azimuthal flow perturbationcauses these particles to rotate due to shear and align at an anglewith respect to the radial direction, leading to a net shear nematicalignment Dry which is strongest near the wall, as well as anazimuthal polarization in the direction opposite to the fluid flow;(iii) hydrodynamic disturbances induced by the swimming activityin the nematically-aligned region tend to reinforce the flowperturbation in the case of pushers via the active forcing termfy = ar"2qr(r

2Dry) in the y-momentum equation. In particular, it canbe checked both in theory and simulations that suspensions ofpullers for which a4 0 are always stable and only exhibit phase I.A more quantitative comparison between theory and simulationsis shown in Fig. 5(c), where the marginal stability curves for thefirst unstable mode for different values of L are found to matchthe numerical transition from phase I to phase II. Fig. 5(c) alsoshows the influence of the parameter L: increasing its value hasa stabilizing effect due to diffusion, which smoothes out thewall accumulation layer responsible for driving the flow.

The low- and high-Peclet limits for the marginal stability ofthe equilibrium state can also be characterized more preciselyas illustrated in Fig. 6. In weakly confined systems (Pes - 0),the marginal value of a for instability becomes independent ofsystem size as shown by the constant asymptote in Fig. 5(c): inthis case, the instability is primarily driven by processes insidethe accumulation layer and an approximate expression for thecritical value of a is derived in Appendix B as

a0c % "32L

1þ 2Las Pes ! 0; (16)

which matches the numerical solution of the linear stabilityproblem excellently, especially for small values of L, as shownin Fig. 6(a). In the limit of strong confinement (Pes - N), weexpect translational diffusion to be the main stabilizing factor,which suggests an asymptote of the form

aNc E "bLPes2 as Pes - N, (17)

where b is an unknown constant. Plotting (a " a0c)/L as a

function of Pes in Fig. 6(b) indeed collapses all the marginal

stability curves onto a single power-law consistent with eqn (17),where the fitting parameter b is found to depend very weakly onL and to asymptote to bE 37.5 at high values of L. A compositeapproximation for the marginal stability curve can thereforebe written as

ac % "32L

1þ 2L" 37:5LPes2; (18)

which provides an excellent fit to our numerical data over a widerange of parameter values. The high-Pes asymptote of eqn (17)can also be used to define a critical domain diameter for theemergence a double vortex in strong confinement:

Dc % 5

ffiffiffiffiffiffiffiffiffi6mdtns0

s

: (19)

Interestingly, this critical domain size only depends upon dt anddoes not involve dr. The scaling with number density Dc B n"1/2

also differs from the scaling of n"1 for the critical system sizefor the onset of collective motion in bulk systems.65

Phase III: turbulent swirling state. As the level of activity keepsincreasing and confinement is decreased, phase II becomesunstable itself leading to phase III, which is an unsteady chaoticstate analogous to that observed in unbounded systems.20,55

There is no clear structure to the flow in this case, which is

Fig. 6 Analytical prediction of the marginal stability curve. (a) Low-Pes

asymptote a0c for the critical value of the activity parameter leading to

instability of the equilibrium state: the plot compares the theoreticalprediction of eqn (16) to the full numerical solution of the linear stabilityproblem (LSA). (b) High-Pes asymptote: the plot shows (a " a0

c)/L as afunction of Pes and confirms the scaling prediction of eqn (17) with fittingparameter b E 37.5.

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instead characterized by local jets and vortices driven by activestresses. Note that the transition to phase III is not predicted byour linear theory, which can only account for axisymmetricdisturbances. Based on previous analyses of unbounded activeturbulence,20–22 we hypothesize that the transition nonethelessresults from a linear instability of the double vortex of phase IIto two-dimensional disturbances, though a more detailed theo-retical analysis remains to be done in this case.

3.2 Periodic channels: circular annuli

We now turn our attention to the case of periodic channels andfirst focus on annulus geometries in which the two boundariesare concentric circles. We use the channel width as the char-acteristic length scale H, and introduce an additional para-meter as the dimensionless inner radius rmin = Rmin/H; detailsof the geometry are shown in Fig. 7. The phenomenology in thiscase is illustrated in Fig. 8, where four distinct regimes areobserved: an axisymmetric equilibrium state with no fluid flow(phase I), an axisymmetric spontaneously flowing state with netfluid pumping (phase II), a spontaneously flowing state withnet fluid pumping and traveling density waves (phase III), and aturbulent-like chaotic state (phase IV). With the exception ofphase III (traveling waves), these regimes are qualitativelysimilar to those found in circular disks. Transitions between

the different states also show similar trends in the (a,Pes)-plane,as depicted in the phase diagram of Fig. 9(a) where phase IIIoccupies a thin region between phases II and IV. Transitionsare also characterized in Fig. 9(b), where the absolute value of themean azimuthal particle velocity | %Vy| is plotted as a function of|a| for different Peclet numbers. As expected, the transition fromequilibrium to net pumping is accompanied by a bifurcationfrom zero to a finite flow rate; the flow rate initially increaseswith activity up to the point when traveling waves appear, afterwhich it start decreasing to reach zero in the chaotic state. Wenow discuss the various regimes in more detail.

Phase I: equilibrium state with no flow. The equilibriumstate shown in Fig. 8(a) is very similar to that found in circulardomains, and is characterized by particle accumulation andwall normal polarization at the outer boundary. The particledistribution at the inner boundary, however, shows a subtledependence on parameter values and can either show accumu-lation (with mr o 0) or depletion (with mr 4 0). This isillustrated in Fig. 10(a) and (b), where we plot radial concen-tration and polarization profiles across the gap for differentdimensional channel widths, (i.e. different values of Pes andrmin at a fixed ratio of Pes/rmin, that we set arbitrarily to 0.5): inwide channels (small Pes) accumulation occurs at both bound-aries, but a depletion is observed in narrow channels (large Pes)in which case attraction by the outer boundary dominates dueto curvature effects and propagates across the entire gap due todiffusion in spite of the presence of the inner wall. Depletionat the inner boundary occurs in strongly diffusive systems orunder strong confinement, where both the concentration andpolarization profiles become linear across the gap. The transitionbetween the two types of distributions is captured in Fig. 10(c)in terms of Pes and rmin for L = 0.5: as rmin decreases for a fixedgap width, the curvature of the outer boundary becomes morepositive while that of the inner boundary becomes morenegative, leading to a higher symmetry breaking between wallsthus fostering depletion at the inner wall. The same transitionis captured in Fig. 10(d) for L = 1, where particle attractionby the outer wall is yet stronger as translational diffusionis enhanced.

Fig. 7 Annulus (left) and racetrack (right) geometries. The annulus ischaracterized by its dimensional inner radius Rmin and width H. Theracetrack has two straight sections of length L and two semi-circularsections whose inner radius is Rmin. The wall-to-wall distance is H. In bothgeometries, the gap width H is chosen as the characteristic length scale fornon-dimensionalization.

Fig. 8 Concentration profiles (colors) and streamlines of the net particle velocity showing four distinct regimes in a circular annulus: (a) equilibrium basestate with no flow (phase I); (b) axisymmetric azimuthal flow with net pumping (phase II); (c) azimuthal flow with net pumping and traveling waves (phaseIII); and (d) turbulent swirling state (phase IV). Results shown are for Pes = 0.5 and L = 0.5. Also see ESI† for a movie showing the dynamics in each case.

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Phase II: spontaneous flow with net pumping. As activity isincreased or confinement is decreased, a first transition to anaxisymmetric flowing state with net fluid pumping occurs(phase II). As previously seen in Fig. 9(b), the mean azimuthalvelocity | %Vy| is non-zero in this regime and increases mono-tonically with the level of activity up to the point where travelingwaves appear (phase III below). The transition to unidirectionalflow is similar to that reported in both bacterial42 and sperm43

suspensions. It was also predicted in a few previous theoreticaland numerical models,44,45 though these typically imposedanchoring boundary conditions on the nematic order para-meter, which are not appropriate to describe suspensions ofswimmers such as bacteria.

As in the case of the disk, the transition to spontaneous flowcan be understood as a linear instability of the equilibriumbase state (phase I) as analyzed more precisely in Fig. 11. Hereagain, an infinite series of unstable modes exists, which involveincreasingly complex azimuthal flow fields with alternatinglayers rotating in opposite directions, for which we show themarginal stability curves and profiles of the net particle velocityin Fig. 11(a) and (b). The first unstable mode, which causes the

strongest pumping, is typically observed in simulations, thoughhigher modes are also seen on rare occasions. As shown inFig. 11(b), upstream swimming generally occurs near the annulusboundaries; this is always true of the outer boundary, though it is insome cases suppressed near the inner boundary when accumulationdoes not occur there as explained in Fig. 10. The dependence of thetransition to net pumping on the inner radius rmin is illustrated inFig. 11(b): the marginal stability curve varies only weakly with rmin

and has a non-monotonic dependence, which again reflects thechange in the nature of the equilibrium base state found in Fig. 10.In very large annuli (rmin - N), the effect of boundary curvaturebecomes negligible locally and the marginal stability curveasymptotes to that for a straight channel.

Phase III: spontaneous flow with traveling waves. At yethigher levels of activity, net pumping persists but the azimuthalsymmetry of the flow is lost and periodic traveling densitywaves appear as shown in Fig. 8(c) as well as Fig. 12 below.Such waves were previously observed in experiments on bothbacterial42 and sperm suspensions,43 and were also predictedby polar active fluid theories.46,47 As illustrated in Fig. 9(b), theaverage azimuthal velocity | %Vy| in this regime systematicallydecreases with activity level, as more and more of the motiontakes place in the radial direction. As for the other steady statesdiscussed previously, this one was found to be stable over longtime intervals. A full characterization of these waves is beyondthe scope of the present work. As either |a| is increased or Pes

decreased, the waves become more intense up to the pointwhen the chaotic state of phase IV emerges; this state is similarto phase III observed above in circular domains.

Fig. 9 Flow transitions in circular annuli. (a) Phase diagram in the (a,Pes)plane for L = 0.5 and rmin = 0.5, showing the transitions between phases I,II, III, and IV. The black curve shows the marginal stability for theequilibrium state of phase I as predicted by a linear stability analysis. (b)Mean azimuthal particle velocity | %Vy| as a function of activity |a| for threedifferent values of Pes.

Fig. 10 Equilibrium distribution (phase I) inside an annulus. (a) Concen-tration and (b) polarization profiles across the gap for different values ofPes, for L = 0.5 and rmin = 2Pes, which is equivalent in dimensional termsto varying the gap width H while keeping the inner radius Rmin constant.(c and d) Transition from accumulation to depletion at r = rmin in the(Pes,rmin)-plane for L = 0.5 and 1, respectively.

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3.3 Periodic channels: racetracks

For direct comparison with the experiments of Wioland et al.,42 weconsider as a final example racetrack geometries composed ofstraight sections of length L connected by two half-annuli as shownin Fig. 7. The characteristic length scale for non-dimensionalizationis still taken to be the channel width H. We focus here on thedynamics in the straight sections; the various flow regimes in thiscase echo those found in circular annuli and are illustrated inFig. 12(a) and (b), showing instantaneous flow patterns andcorresponding mean velocity profiles. A transition to net pumpingis first observed upon increasing channel width at a fixed value of a,followed by the appearance of traveling waves. As the waves becomestronger the flow takes the form of alternating counter-rotatingvortices and eventually destabilizes into chaos. This phenomenol-ogy is identical to that observed in the experiments42 (see Fig. 4 ofthat reference). The mean velocity profiles are also consistent withthe theoretically predicted unstable linear eigenmodes shown inFig. 12(c) for the same parameter values. The last row in Fig. 12demonstrates the possibility of a more complex flow regimethat is only rarely observed: here the flow has a complexunsteady structure, but the mean velocity profile highlightstwo counter-flowing streams near the top and bottom walls andresembles the second unstable linear mode.

We finish by describing the relationship between the onsetof spontaneous pumping in confinement and the effectiverheology of the system. In recent work, Alonso-Matilla et al.51

calculated the effective relative viscosity Zr in a dilute activesuspension confined in a planar channel and subject to anapplied pressure-driven flow, where Zr is defined as the ratio ofthe flow rate in pure fluid by that in the presence of swimmers ata given pressure gradient. In agreement with previous experi-ments in the same geometry,48 suspensions of pushers werefound to enhance the flow, i.e. decrease the effective viscosity ofthe suspension as a result of activity. This effect was found to bethe strongest in the limit of vanishing flow strength, and thezero-flow-rate viscosity Z0

r was further found to decrease withincreasing |a|, until it eventually reaches zero suggesting anapparent transition to superfluidity. The exact dependence of Z0

r

with activity is plotted in Fig. 13(a), where it is found that thevalue of |a| for which Z0

r reaches zero coincides precisely with theprediction of the linear theory for the marginal stability of theequilibrium state and onset of spontaneous flow. This providesan additional interpretation for the transition to pumping as aconsequence of apparent superfluidity: as the resistance of thesystem to flow is effectively zero, a small perturbation in the fluidvelocity can amplify at no cost leading to unidirectional flow.In the flowing state, the input of mechanical energy by theswimmers exactly balances viscous dissipation in the solvent.This interpretation is further borne out by Fig. 13(b), which plotsthe mean flow rate in the spontaneous flow regime as a function ofthe length L of the straight section of the racetrack. Remarkably,we observe that the flow rate in the pumping regime is completelyindependent of channel length, another signature of an effectivelyfrictionless flow.

4 Concluding remarksWe have used a combination of numerical simulations andtheory to explore the effect of confinement and geometry onthe onset and structure of spontaneous flows in semi-dilutesuspensions of active swimming microorganisms. A mean-fieldtheory based on the coupled Smoluchowski and Navier–Stokesequations was used to describe the dynamical evolution ofswimmer configurations. We solved these governing equationsnumerically in two dimensions and compared our numericalresults with predictions from a linear stability analysis. Ourresults agree well with prior experimental studies and were ableto capture and explain the spontaneous directed motionsarising in pusher suspensions as a result of hydrodynamicinteractions alone. The case of puller particles, for which a 4 0,was not considered here as instabilities and spontaneous flowswere never found to arise in our simulations.

We first analyzed the dynamics of swimmers in circular disks,where three distinct flow regimes were found depending onthe level of activity and degree of confinement: equilibrium withno flow, a steady double vortex, and a swirling chaotic state. Theequilibrium state manifests at low levels of activity or strongconfinement, where the spatial and orientational distribution ofparticles is axisymmetric and the net disturbance flow generatedby the swimmers vanishes. Particles accumulate at the boundariesand on average are radially polarized towards the wall, reaching

Fig. 11 Linear stability of the equilibrium base state (phase I) in a circularannulus. (a) Marginal stability curves in the (a,Pe)-plane for the firstunstable modes at L = 0.5 and rmin = 0.5. (b) Unstable eigenmodes forthe net particle velocity Vy for Pes = 0.5, L = 0.5, and rmin = 0.5 at the onsetof instability. (c) Marginal stability curves for the first unstable mode in the(a,rmin) plane for different values of Pes and for L = 0.5.

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their maximum polarization at the boundaries. Increasingthe level of activity or decreasing confinement destabilizesthe system into the double vortex state, and a mechanismbased on the shear alignment of the swimmers in the distur-bance flow they generate inside the accumulation layerwas uncovered. By the same shear alignment mechanism,swimmers were in fact shown to orient against the flowthroughout the domain allowing them to swim upstream nearthe boundary, thus leading to the double vortex structure reportedin experiments where bacterial velocities were measured. A furtherincrease in the level of activity or a decrease in confinementoriginates a second transition to a turbulent-like chaotic stateanalogous to that observed in bulk systems.

We then turned our focus to swimmer dynamics in periodicgeometries. Our simulations in circular annuli captured fourflow regimes quite similar to those found in circular disks: anequilibrium state with no flow, an axisymmetric state with net

fluid pumping, the emergence of traveling density waves, followedby a chaotic state. The transitions between regimes were againfound to be governed by the level of activity and degree ofconfinement, with only a weak dependence on the inner radiusdimension. Similar transitions were also observed in periodicracetracks, where we were able to connect the onset of sponta-neous pumping with the effective rheology of the suspension.Specifically, the transition to net pumping was shown to occur atthe same level of activity at which the zero-shear-rate viscositybecomes zero in a pressure-driven planar channel flow, suggestingthat spontaneous flows in confinement are in fact a consequenceof the apparent superfluidity of the system. This conclusion wasfurther supported by observing that the net flow rate is indepen-dent of channel length in periodic racetracks.

Our numerical and theoretical results both underscore thesubtle interplay between confinement, geometry, and activity insemi-dilute active suspensions. While this study focused on

Fig. 12 (a) Concentration profiles and streamlines of the net particle velocity in straight sections of periodic racetracks, showing four different flowregimes (from top to bottom). (b) Average net velocity profiles across the gap in each case. (c) Unstable linear eigenmodes for these parameter values,obtained in straight channels. Results shown are for a = "60 and L = 0.5. Also see ESI† for a movie showing the dynamics in each case.

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fairly simple geometries previously considered in experiments,we anticipate that a wealth of more complex flow regimes mightarise in other types of geometries. Continuum modeling asperformed in this work proves to be a valuable tool for theunderstanding and prediction of these flows and could alsoplay a useful role in the design of microfluidic devices forbioengineering applications involving bacteria.

Appendix A: analytical solutions forequilibrium statesIf the nematic order tensor is neglected (D = 0), which is a goodapproximation at equilibrium as shown by the full numericalsolution, simple closed-form analytical solutions can be derivedfor the radial concentration and polarization profiles which weprovide here. In axisymmetric geometries, the steady governingequations for c(r) and mr(r) obtained by setting the Fc and Fm ineqn (9) and (10) in the absence of flow simplify to:

" d

drrmrð Þ þ LPes

d

drrdc

dr

" #¼ 0; (20)

"Pes2

dc

drþ LPes2

d

dr

1

r

d

drrmrð Þ

* +"mr ¼ 0: (21)

In a circular disk, the boundary conditions at r = 1 are:

"mr þ LPesdc

dr¼ 0; (22)

"c2þ LPes

dmr

dr¼ 0; (23)

and we also require that the solution remain bounded at r = 0and satisfy the normalization

ð1

0cðrÞrdr ¼ 1

2: (24)

Integrating eqn (20) once easily shows that the boundarycondition of eqn (22) in fact applies everywhere across thegap and expresses the local balance between swimming anddiffusive fluxes. Inserting eqn (22) into eqn (21) and manip-ulating then provides a modified Bessel equation for mr(r):

r2d2mr

dr2þ r

dmr

dr" 1þ O2r2$ %

mr ¼ 0; (25)

where

O2 ¼ 1

Pes2L1þ 1

2L

" #: (26)

After applying boundary conditions, the concentration andpolarization profiles are obtained as

c(r) = A1 + A2I0(Or), mr(r) = A2LPesOI1(Or), (27)

where the constants A1 and A2 are expressed in terms ofincomplete Bessel functions as

A1 ¼ 1" 2

OI1ðOÞA2; (28)

A2 ¼ L2Pes2O2 " 1

$ %I0ðOÞ þ

2

OI1ðOÞ þ L2Pes

2O2I2ðOÞ* +"1

:

(29)

The expression for c(r) in eqn (27) is identical to that previouslyobtained by Yan & Brady.60 The solution inside an annulus isalso easily obtained by applying boundary conditions of eqn (22)and (23) at both walls r = rmin and rmin + 1 but is omitted here forbrevity. The solution in a straight channel was also previouslycalculated by Ezhilan & Saintillan.30

Appendix B: low- and high-Pes

marginal stability limits in a circular diskThe stability analysis is performed by perturbing the governingequations about the equilibrium state, which we denoteby (c0,m0,D0). We focus here on the marginal stability, forwhich the growth rate is set to zero. In large domains, theeffect of boundary curvature on the structure of the accumula-tion layer is negligible, which prompts us to use Cartesiancoordinates. Upon linearization of the equations, we arrive at a

Fig. 13 (a) Zero-flow-rate relative velocity Zr as a function of activity |a| ina suspension of pushers confined between two flat plates, obtained usingthe model of Alonso-Matilla et al.51 As activity increases, the relativeviscosity decreases and reaches superfluidity at the critical value of |a|for the onset of spontaneous flows as predicted by our linear stabilityanalysis (LSA). This calculation was performed in three dimensions for easeof comparison with the rheological model. (b) Average longitudinal velo-city |V8| as a function of channel length L in periodic racetracks in thespontaneous flow regimes (phases II and III).

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coupled system for the variables mx0 and Dxy

0 (where 0 refersto perturbations):

0 ¼ "PesdDxy

0

dyþ LPes2

d2mx0

dy2" 3

4acDxy

0m0

y "mx0; (30)

0 ¼ "14Pes

dmx0

dyþ LPes2

d2Dxy0

dy2" acDxy

0 c0

4þD0

yy

" #" 4Dxy

0:

(31)

These constitute an eigenvalue problem for (mx0,Dxy

0) withcorresponding eigenvalue ac.

In the low-Pes limit, the dominant balance in eqn (31) isbetween the last two terms, which capture flow alignment androtational diffusion. An estimate for ac can then be obtained bybalancing these two terms and by using the maximum valuesof c0 and D0

yy, which are attained at the walls:

a0c %"4

c0wall4þD0

yy;wall

¼ " 32L1þ 2L

as Pes ! 0; (32)

which indeed agrees with the full numerical solution of theeigenvalue problem as shown in Fig. 6(a).

In the high-Pes limit (strong confinement), the dominantbalance now takes place between the second and third terms ineqn (31), which describe translational diffusion and shearalignment. While the use of Cartesian coordinates is no longerjustified in this case, the form of the equations still suggestsa scaling of the type

aNc E "bLPes2 as Pes - N, (33)

which is again supported by numerical data with b E 37.5 asshown in Fig. 6(b).

AcknowledgementsThe authors thank Barath Ezhilan, Jeremie Palacci, HugoWioland, Zvonimir Dogic, and Michael Shelley for valuablediscussions. Funding from NSF Grants CBET-1532652 and DMS-1463965 is gratefully acknowledged.

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